# Intersection of Cylinder and Plane

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\documentclass{article}
%
% Nombre de archivo: intersection-cylinder-plane.tex
% Description:
% A solid bounded by the following surfaces
% z = 0
% z = \sqrt{x^{2} + y^{2}}
% x^{2} + y^{2} + z^{2} = 1
% is generated. I.e., the intersection of a cylinder and a plane.
%
% Date of last modification: April, 23rd, 2022.
% Author: Efraín Soto Apolinar.
% https://www.aprendematematicas.org.mx/author/efrain-soto-apolinar/instructing-courses/
% According to TikZ.net
%
\usepackage{tikz}
\usetikzlibrary{patterns}
\usepackage{tikz-3dplot}
\usepackage[active,tightpage]{preview}
\PreviewEnvironment{tikzpicture}
\setlength\PreviewBorder{1pt}
%
\begin{document}
%
\tdplotsetmaincoords{60}{110}
\begin{tikzpicture}[tdplot_main_coords,scale=0.75]
\pgfmathsetmacro{\final}{2.0*pi}
% The equation of the circumference
\draw[white] (0,-2.5,0) -- (2.75,0,0) node[blue,midway,below,sloped] {\footnotesize$x^2 + y^2 = 4$};
% Region of integration: circle of radius 2
\draw[thick,fill=yellow,opacity=0.5] plot[domain=0:6.2832,smooth,variable=\t] ({2.0*cos(\t r)},{2.0*sin(\t r)},{0.0});
%%% Coordinate axis
\draw[thick,->] (0,0,0) -- (3.5,0,0) node [below left] {\footnotesize$x$};
\draw[dashed] (0,0,0) -- (-3,0,0);
\draw[thick,->] (0,0,0) -- (0,3.5,0) node [right] {\footnotesize$y$};
\draw[dashed] (0,0,0) -- (0,-3,0);
\draw[thick] (0,0,0) -- (0,0,4);% node [above] {\footnotesize$z$};
% The plane: x + y = 2
\coordinate (A) at (3,2.5,1.5);
\coordinate (B) at (-3,2.5,1.5);
\coordinate (C) at (-3,-2.5,6.5);
\coordinate (D) at (3,-2.5,6.5);
% The Cylinder
\foreach \angulo in {0,0.01,...,\final}{
\pgfmathparse{2.0*cos(\angulo r)}
\pgfmathsetmacro{\px}{\pgfmathresult}
\pgfmathparse{2.0*sin(\angulo r)}
\pgfmathsetmacro{\py}{\pgfmathresult}
\draw[cyan,opacity=0.5] (\px,\py,0) -- (\px,\py,4.0-\py);
}
% Trace (intersection of the plane and the cylinder)
\draw[blue,thick] plot[domain=0:6.2832,smooth,variable=\t] ({2.0*cos(\t r)},{2.0*sin(\t r)},{4.0-2.0*sin(\t r)});
% Circumferences bounding the cylinder
\draw[blue,thick,opacity=0.5] plot[domain=0:6.2831853,smooth,variable=\t] ({2.0*cos(\t r)},{2.0*sin(\t r)},{4.0-2*sin(\t r)});
\draw[blue,thick,opacity=0.5] plot[domain=0:6.2831853,smooth,variable=\t] ({2.0*cos(\t r)},{2.0*sin(\t r)},{0.0});
% The plane
\draw[white] (C) -- (B) node[red,above,sloped,midway]{\footnotesize$y + z = 4$};
\draw[red,dash dot] (A) -- (B) -- (C)	 -- (D) -- (A);
\fill[pattern color=pink,pattern=north east lines] (A) -- (B) -- (C)	 -- (D) -- (A);
\fill[yellow,opacity=0.35] plot[domain=0:6.2832,smooth,variable=\t] ({2.0*cos(\t r)},{2.0*sin(\t r)},{4.0-2.0*sin(\t r)});
% Tz axis (last part)
\draw[thick,->] (0,0,4) -- (0,0,8) node [above] {\footnotesize$z$};
\end{tikzpicture}
\end{document}